Truth tables help determine the input combination that will yield a certain output value; This is useful when we want to translate a given truth table into a boolean equation that can be much easily manipulated and simplified before actually building a circuit, hopefully making the wiring eaasier and cheaper by using less components.
There are two complementary terms that we use to accomplish this: Minterms and Maxterms.
A minterm represents each row of the truth table that has an output of 1. To translate a truth table row into the corresponding minterm we AND (or multiply) each of the terms at the input, inverting (applying a NOT operator) to each variable whose state for that particular row happens to be zero.
For example, the three input truth table:
A B C Z
0 0 0 0
0 0 1 0
0 1 0 1 <
0 1 1 0
1 0 0 1 <
1 0 1 0
1 1 0 0
1 1 1 1 <
The rows marked with an arrow represent the minterms of the table. The equation for this table would be
Z = (~A * B * ~C) + (A * ~B * ~C) + (A * B * C)
Note that all three minters go together in the same equation, since any of them can trigger an output 1 (if the first is true [1] OR [+] the second OR the third, the output is also true [1]).
Minterms are also called the sum of products representation because of the way they end up arranged in the equation.
Maxterms are the complementary operation of minterms. Maxterms are obtained from the rows that have a zero in them as output. Using the above example, all the rows not marked with an arrow are the table's maxterms.
To translate from the table to a boolean equation we OR (sum) each of the terms acting as input, applying a NOT operation to any input that happens to be a 1 for that particular row. Notice how the operation (AND for minterms, OR for maxterms) and the criteria for negation (when the input variable is 1 for minterms and when it is 0 for maxterms) are opposite of each other.
The equation in Maxterms for the example would be
Z = (A + B +C)*(A + B + ~C)*(A + ~B + ~C)*(~A + B + C)*(~A + ~B + C)
For this particular table, you can see that the equation in maxterms has more of them, this is because there are more 0's as output than 1's, and since each row having them is one term in the equation, the more there are the more terms the resulting equation will have.
The maxterm representation is also called a product of sums, because of the way they are arranged.
They are arranged in such a way because if any of them is 0, then the output should be 0 as well, even if the other terms are 1. This is because if one of the terms is 0, then it means that the combination of inputs matches one of the rows of the table that results in a 0 output, and no matter what the other terms are (any number multiplied by 0 is 0), the output should be 0 in order for the equation to work the same as what's specified in the table.
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